Factor the following expression completely. x^(4)-4x^(3)-12x^(2)-4x^(2)+16 x+48 Answer:

Combine Like Terms: First, let's combine like terms in the expression. x^4 - 4x^3 - 12x^2 - 4x^2 + 16x + 48 = x^4 - 4x^3 - (12x^2 + 4x^2) + 16x + 48 = x^4 - 4x^3 - 16x^2 + 16x + 48Find Common Factors: Now, we look for common factors in groups of terms. We can group the terms as follows: (x^4 - 4x^3) and (-16x^2 + 16x + 48). Let's factor out the greatest common factor from each group. For the first group, the greatest common factor is x^3. For the second group, the greatest common factor is 16. So we have: x^3(x - 4) - 16(x^2 - x - 3)Factor Quadratic Expression: Next, we factor the quadratic expression in the second group. The quadratic x^2 - x - 3 can be factored into (x - 3)(x + 1). So the expression becomes: x^3(x - 4) - 16(x - 3)(x + 1)Check for Common Factor: Now, we look for a common factor in both groups. We notice that there is no common factor between x^3(x - 4) and -16(x - 3)(x + 1). Therefore, the expression is already factored completely.

Home

Math Problems

Precalculus

Operations with rational exponents

Full solution

Q. Factor the following expression completely.

\newline

x^{4}-4 x^{3}-12 x^{2}-4 x^{2}+16 x+48

\newline

Answer:

Combine Like Terms: First, let's combine like terms in the expression. $\newline$ $x^4 - 4x^3 - 12x^2 - 4x^2 + 16x + 48$ $\newline$ = $x^4 - 4x^3 - (12x^2 + 4x^2) + 16x + 48$ $\newline$ = $x^4 - 4x^3 - 16x^2 + 16x + 48$
Find Common Factors: Now, we look for common factors in groups of terms. $\newline$ We can group the terms as follows: $(x^4 - 4x^3)$ and $(-16x^2 + 16x + 48)$ . $\newline$ Let's factor out the greatest common factor from each group. $\newline$ For the first group, the greatest common factor is $x^3$ . $\newline$ For the second group, the greatest common factor is $16$ . $\newline$ So we have: $\newline$ $x^3(x - 4) - 16(x^2 - x - 3)$
Factor Quadratic Expression: Next, we factor the quadratic expression in the second group. $\newline$ The quadratic $x^2 - x - 3$ can be factored into $(x - 3)(x + 1)$ . $\newline$ So the expression becomes: $\newline$ $x^3(x - 4) - 16(x - 3)(x + 1)$
Check for Common Factor: Now, we look for a common factor in both groups. $\newline$ We notice that there is no common factor between $x^3(x - 4)$ and $-16(x - 3)(x + 1)$ . $\newline$ Therefore, the expression is already factored completely.